Question 12846: A Space-Wide Certificate Is Not a List of Successful Sequences

Original question and givens

Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 14, 15], [9, 14, 0, 17], [10, 15, 17, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=sampled_sequences. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.

• G001 — Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 14, 15], [9, 14, 0, 17], [10, 15, 17, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=sampled_sequences. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-complete-metric-space

Step-by-step derivation

carrier Fin(4);complete nonnegative integer metric candidate=[[0, 7, 9, 10], [7, 0, 14, 15], [9, 14, 0, 17], [10, 15, 17, 0]]

zero iff equal=Yes;symmetric=Yes

endpoints=[0, 0];direct distance=0;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[0, 14, 18, 20];direct<=every two-leg=Yes

endpoints=[0, 1];direct distance=7;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[7, 7, 23, 25];direct<=every two-leg=Yes

endpoints=[0, 2];direct distance=9;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[9, 21, 9, 27];direct<=every two-leg=Yes

endpoints=[0, 3];direct distance=10;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[10, 22, 26, 10];direct<=every two-leg=Yes

endpoints=[1, 0];direct distance=7;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[7, 7, 23, 25];direct<=every two-leg=Yes

endpoints=[1, 1];direct distance=0;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[14, 0, 28, 30];direct<=every two-leg=Yes

endpoints=[1, 2];direct distance=14;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[16, 14, 14, 32];direct<=every two-leg=Yes

endpoints=[1, 3];direct distance=15;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[17, 15, 31, 15];direct<=every two-leg=Yes

endpoints=[2, 0];direct distance=9;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[9, 21, 9, 27];direct<=every two-leg=Yes

endpoints=[2, 1];direct distance=14;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[16, 14, 14, 32];direct<=every two-leg=Yes

endpoints=[2, 2];direct distance=0;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[18, 28, 0, 34];direct<=every two-leg=Yes

endpoints=[2, 3];direct distance=17;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[19, 29, 17, 17];direct<=every two-leg=Yes

endpoints=[3, 0];direct distance=10;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[10, 22, 26, 10];direct<=every two-leg=Yes

endpoints=[3, 1];direct distance=15;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[17, 15, 31, 15];direct<=every two-leg=Yes

endpoints=[3, 2];direct distance=17;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[19, 29, 17, 17];direct<=every two-leg=Yes

endpoints=[3, 3];direct distance=0;all intermediate labels=[0, 1, 2, 3]

all two-leg distances=[20, 30, 34, 0];direct<=every two-leg=Yes

all off-diagonal positive distances=[7, 9, 10, 7, 14, 15, 9, 14, 17, 10, 15, 17]

minimum positive distance delta=7>0

qualified finite metric;delta=7;selected evidence mode=sampled_sequences

selected evidence sufficient=No;space completeness is not denied by sample insufficiency

Reverse checks

• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(4, 7, sampled_sequences, No, sampled_sequences_insufficient)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.